Simplify and expand the following expression: $ \dfrac{4y}{4y - 6}+\dfrac{4y - 9}{5y - 1} $
Solution: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(4y - 6)(5y - 1)$ Multiply the first term by $\dfrac{5y - 1}{5y - 1}$ $ \begin{align*} \dfrac{4y}{4y - 6} \times \dfrac{5y - 1}{5y - 1} & = \dfrac{(4y)(5y - 1)}{(4y - 6)(5y - 1)} \\ & = \dfrac{20y^2 - 4y}{(4y - 6)(5y - 1)}\end{align*} $ Multiply the second term by $\dfrac{4y - 6}{4y - 6}$ $ \begin{align*} \dfrac{4y - 9}{5y - 1} \times \dfrac{4y - 6}{4y - 6} & = \dfrac{(4y - 9)(4y - 6)}{(5y - 1)(4y - 6)} \\ & = \dfrac{16y^2 - 60y + 54}{(5y - 1)(4y - 6)}\end{align*} $ Now we have: $ = \dfrac{20y^2 - 4y}{(4y - 6)(5y - 1)} + \dfrac{16y^2 - 60y + 54}{(5y - 1)(4y - 6)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{20y^2 - 4y + 16y^2 - 60y + 54}{(4y - 6)(5y - 1)} $ $ = \dfrac{36y^2 - 64y + 54}{(4y - 6)(5y - 1)}$ Expand the denominator: $ = \dfrac{36y^2 - 64y + 54}{20y^2 - 34y + 6}$ Simplify: $ = \dfrac{18y^2 - 32y + 27}{10y^2 - 17y + 3}$